Limiting dynamics for stochastic Navier-Stokes equations on expanding unbounded domains

We study the limiting dynamics of stochastic non-autonomous Navier-Stokes equations defined on a sequence of expanding domains, where the largest is an unbounded Poincaré domain. We prove the upper semi-continuity of the null-expansion of the corresponding random attractor when the bounded domain is expanded to the unbounded domain. To do this, we expand each random dynamical system (cocycle) and then prove the expanding cocycle converges to the cocycle on the unbounded domain. By generalizing the famous energy equation method, we prove that the sequence of expanding cocycles is weakly equi-continuous and strongly equi-asymptotically compact, which lead to the upper semi-continuity of attractors.

Keywords

Navier-Stokes equation, random attractor, upper semi-continuity, expanding cocycle, expanding domain, energy method

2010 Mathematics Subject Classification

35B40, 35B41, 60H15

The full text of this article is unavailable through your IP address: 18.223.210.196

Received 7 September 2018

Accepted 12 October 2023

Published 12 July 2024